Problem 75 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int \frac{4}{1+x^{2}} d x $$

Step-by-Step Solution

Answer

The indefinite integral is \( 4 \arctan(x) + C \).

Step 1: Recognize the Integral Form

Recognize that the integral \( \int \frac{4}{1+x^2} \ dx \) is similar to the standard integral \( \int \frac{1}{1+x^2} \ dx = \arctan(x) + C \), where \( C \) is the constant of integration.

Step 2: Simplify the Integral

Factor out the constant 4 from the integral: \( 4 \int \frac{1}{1+x^2} \ dx \). This simplifies the problem into a constant multiplied by a standard form integral.

Step 3: Integrate the Function

Use the standard integral result to integrate: \( 4 \times \arctan(x) \). This gives the result of the integral as \( 4 \arctan(x) + C \).

Step 4: State the Solution

Write out the complete solution, including the constant of integration: \( \int \frac{4}{1+x^2} \ dx = 4 \arctan(x) + C \).

Key Concepts

Standard Integral Forms

In calculus, standard integral forms serve as a reference to solve a broad range of integrals quickly and accurately. One of the most common forms is the integral of \[ \int \frac{1}{1+x^2} \, dx = \arctan(x) + C \]where \( C \) is the constant of integration.
These forms allow us to recognize patterns in integrals and apply predefined results.

• They save time and effort by avoiding the need for lengthy computations.
• Patterns like \( \int \frac{1}{x} \, dx = \ln|x| + C \) or the power rules \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) (for \( n eq -1 \)) help quickly deduce answers.

In the given problem, recognizing how the integral of \( \frac{4}{1+x^2} \) is a multiple of the arctangent form allows us to simplify and solve the problem efficiently, leading to the final solution of \( 4 \arctan(x) + C \).

Arctangent Function

The arctangent function, denoted as \( \arctan(x) \), is the inverse of the tangent function.
Understanding this function is crucial in calculus as it appears frequently in integration, especially with forms like \( \int \frac{1}{1+x^2} \, dx \).

• The arctangent function helps in finding the angle \( \theta \) given its tangent: \( \theta = \arctan\left( \frac{opposite}{adjacent} \right) \).
• Its range is \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \), which means it returns values between these limits.

In integration, knowing the derivative forms, such as \( \frac{d}{dx} \arctan(x) = \frac{1}{1+x^2} \), helps in reversing the process to obtain the integral.
Thus, applying the standard integral form to \( \int \frac{1}{1+x^2} \, dx \), we directly use the arctangent function, greatly simplifying problem-solving.

Constant of Integration

In indefinite integrals, the constant of integration, denoted by \( C \), is essential.
Every indefinite integral includes this constant to account for all possible antiderivatives.

• Consider if two functions, \( F(x) \) and \( F(x) + C \), differ by a constant \( C \), differentiating them gives the same derivative.
• This reflects that there are infinitely many antiderivatives for a function, differing only by this constant.

Specifically, for problems like \( \int \frac{4}{1+x^2} \), although we integrate to obtain \( 4 \arctan(x) \), adding \( C \) ensures all solutions are considered.
Thus, it's crucial never to omit \( C \) in indefinite integrals because it represents the family of functions generated by the antiderivative.